In mathematics and logic , a (FINITARY ) BOOLEAN FUNCTION (or
switching function) is a function of the form _ƒ_ : B_k_ → B, where
B = {0, 1} is a _
Boolean domain _ and _k_ is a non-negative integer
called the arity of the function. In the case where _k_ = 0, the
"function" is essentially a constant element of B.

Every _k_-ary
Boolean function can be expressed as a propositional
formula in _k_ variables _x_1, …, _x__k_, and two propositional
formulas are logically equivalent if and only if they express the same
Boolean function. There are 22_k_ _k_-ary functions for every _k_.

BOOLEAN FUNCTIONS IN APPLICATIONS

A
Boolean function describes how to determine a Boolean value output
based on some logical calculation from Boolean inputs. Such functions
play a basic role in questions of complexity theory as well as the
design of circuits and chips for digital computers . The properties of
Boolean functions play a critical role in cryptography , particularly
in the design of symmetric key algorithms (see substitution box ).

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